Componentwise asymptotic stability of linear constant dynamical systems

This note deals with a special type of asymptotic stability, namely componentwise asymptotic stability with respect to the vectorgamma(t)(CWASγ) of systemS: dot{x} = Ax + Bu, t geq 0, wheregamma(t) > 0(componentwise inequality) andgamma(t) rightarrow 0ast rightarrow + infty.Sis CWASγ if for eacht_{0} geq 0and for each|x(t_{0})| leq gamma (t_{0}) (|x (t_{0})|with the components|x_{i}(t_{0})|the free response ofSsatisfies|x(t)| leq gamma (t)for eacht geq t_{0}. Forgamma(t){underline { underline delta} } alphae^{-beta t}, t geq 0, withalpha > 0andbeta > 0(scalar), the CWEAS (E= exponential) may be defined.Sis CWAS γ (CWEAS) if and only ifdot{gamma}(t) geq bar{A}gamma(t), t geq 0 (bar{A}alpha < 0); A {underline { underline delta} } (a_{ij})andbar{A}has the elements aijand|a_{ij}|, i neq j. These results may be used in order to evaluate in a more detailed manner the dynamical behavior ofSas well as to stabilizeScomponentwise by a suitable linear state feedback.     

On the fuel-optimal singular control of nonlinear second-order systems

Given a body subject to quadratic drag forces so that the positiony(t)and the applied control thrustu(t)are related byddot{y}(t)+adot{y}(t)|dot{y}(t)| = u(t), |u(t)| leq 1, the controlu(t)is found which forces the body to a desired position, and stops it there, and which minimizes the costJ=intliminf{0} limsup{T}{k + |u(t)|}dt. The response timeTis not fixed,k > 0, and|u(t)|is proportional to the rate of flow of fuel. Repeated use of the necessary conditions provided by the Maximum Principle results in the optimum feedback system. It is shown that ifkleq 1, then singular controls exist and they are optimal; ifk > 1, then singular controls are not optimal. Techniques for the construction of the various switch curves are given, and extensions of the results to other nonlinear systems are discussed.     

Minimum-energy control of a second-order nonlinear system

This paper establishes the bounded control functionu(t)which minimizes the total energy expended by a submerged vehicle (for propulsion and hotel load) in a rectilinear translation with arbitrary initial velocity, arbitrary displacement, and zero final velocity. The motion of the vehicle is determined by the nonlinear differential equationddot{x}+adot{x}|dot{x}| = u, a > 0. The performance index to be minimized is given byS =int_{0}^{T}(k+udot{x})dt, withTopen andk > 0.The analysis is accomplished with the use of the Pontryagin maximum principle. It is established that singular controls can result whenk leq 2 sqrt{U^{3}/a}.Uis the maximum value of|u(t)|.     

Stability of polynomials under coefficient perturbation

Let the real polynomiala_{0}x^{n} + a_{1}x^{n-1} + ... + a_{n-1}x + a_{n}be stable and let the real numbersb_{k}, c_{k} geq 0, 0 leq k leq n, be given. We present a simple determinant criterion for finding the largestt_{0} geq 0such that the polynomialalpha_{0}x^{n} + alpha_{1}x^{n-1}+ ... +alpha_{n-1}x + alpha_{n}is stable for allalpha_{k} in (a_{k} - b_{k}t_{0}, a_{k} + C_{k}t_{0}) cup {a_{k}}, 0 leq k leq n. Several further observations allow us to reduce the computational cost considerably.     

Application of the describing function technique in a single-loop feedback system with two nonlinearities

A graphic procedure is presented which allows the describing function technique to be extended to a single-loop feedback system with two nonlinearities. The graphic technique is very simple and immediately allows qualitative answers, or quantitative answers subject to the usual errors and restrictions of the describing function technique, to be obtained regarding the presence of limit cycles, regions of stability, instability, etc. The method essentially is as follows. A plot ofG_{1}(jomega) G_{2}(j_omega)in Fig. 1 vs. ω is made, and the point of intersection ofG_{1}(jomega) G_{2}(jomega)with the negative real axis is noted, for example, atG_{1}(jomega^{ast}) G_{2}(jomega^{ast}) =-1/Gamma, Gamma > 0. By plotting |G_{d_{1}}(A1)| vs. A1in the second quadrant, and|G_{d_{2}}(A_{2})|vs. A2in the fourth quadrant, it is possible to plot a curve (relating |G_{d_{1}}| vs. |G_{d_{1}}|) in the first quadrant. If this curve intersects|G_{d_{1}}| |G_{d_{2}}| = Gamma, a limit cycle exists in the system. If no intersection takes place, then no limit cycle exists in the system.     

Time-, fuel-, and energy-optimal control of nonlinear norm-invariant systems

Nonlinear systems of the formdot{X}(t)=g[x(t);t]+u(t), wherex(t), u(t), andg[x(t); t]arenvectors, are examined in this paper. It is shown that ifparellelx(t)parellel = sqrt{x_{1}^{2}(t) + ... + x_{n}^{2}(t)}is constant along trajectories of the homogeneous systemdot{X}(t)=g[x(t); t]and if the controlu(t)is constrained to lie within a sphere of radiusM, i.e.,parellelu(t}parellel leq M, for allt, then the controlu^{ast}(t)= - Mx(t} /parellelx(t)parelleldrives any initial statexito 0 in minimum time and with minimum fuel, where the consumed fuel is measured byint liminf{0} limsup{T}parellel u(t) parelleldt. Moreover, for a given response timeT, the controlutilde(t) = -parellelxiparellel x(t)/T parellel x(t) parelleldrivesxito 0 and minimizes the energy measured byfrac{1}{2}int liminf{0} limsup{T}parellelu(t)parellel^{2}dt. The theory is applied to the problem of reducing the angular velocities of a tumbling asymmetrical space body to zero.     

A characterization of eAtand a constructive proof of the controllability criterion

A differential equation characterizing the functionsalpha_{i}(t), which arise when e^Atis expressed asalpha_{0}(t)I + ... + alpha_{n-1}(t) A^{n-1}, is derived. It is shown that the set of functions{alpha_{i}(t)}is linearly independent over any nonzero interval. Using this fact, a constructive proof is given for the well-known criterion for a linear time-invariant system to be controllable, namely, rank[B|AB| ... |A^{n-1}B] = n.     

A synthesis theory for linear time-varying feedback systems with plant uncertainty

Given a feedback system containing a linear, time-varying (LTV) plant with significant plant uncertainty, it is required that the system response to command and disturbance inputs satisfy specified tolerances over the range of plant uncertainty. The synthesis procedure guarantees the latter satisfied, providing that they are of the following form. Leth(t',tau)be the system response att'= t - taudue to a command inputdelta(t - tau), andh_{tau}(s)=int liminf{0}limsup{infty}h(t',tau)e^-{st'}dt'is the Laplace transform ofh(t',tau). There is given a setM_{tau}(omega)={m_{tau}(omega)} , omega in[0, infty), with the requirement that|h_{tau}(jomega)| in M_{tau}(omega), over the range of plant uncertainty. The disturbance response tolerances are of the same form, in response to a disturbance inputdelta (t- tau). The acceptable response setM_{tau}(omega)can depend on τ. The design emerges with a fixed pair of LTV compensation networks and can be considered applicable to time-domain response tolerances, to the extent that a set of bounds on a time function can be translated into an equivalent set on its frequency response. The design procedure utilizes only time-invariant frequency response concepts and is conceptually easy to follow and implement. At any fixed τ, the time-varying system is converted into an equivalent time-invariant one with plant uncertainty, for which an exact solution is available, with "frozen" time-invariant compensation. Schauder's fixed-point theorem is used to prove the equivalence of the two systems. The ensemble over τ of the time-invariant compensation gives the final required LTV compensation. It is proven that the design is stable and nonresonant for all bounded inputs.     

Improving the predictions of the circle criterion by combining quadratic forms

By using a Lyapunov function which consists of different quadratic forms in various sectors of the (u, (du/dtau)) plane, the prediction of the circle criterion that the null solution of(d^{2}u/dtau^{2}) + 2(du/dtau) + f(tau, u, (du/dtau))cdotp u = 0is asymptotically stable for0 leq alpha < f(cdotp) < beta, withbeta = (sqrt{alpha} + 2)^{2}, is improved tobeta = [{frac{(sqrt{alpha} + 1)^{2} + 1 + sqrt{(sqrt{alpha} + 1)^{4} + 2 (sqrt{alpha} + 1)^{2} + 5}}{2}}^{frac{1}{2}} + 1 ]^{2}.     

On the controllability of discrete linear systems with output feedback

A discrete time linear systemx_{t+1}= Ax_{t} + Bu_{t'}y = Cx_{t'}, with output feedbacku_{t} = G_{t}y_{t'}, call be regarded as a nonlinear system with "control" Gt. Weak sufficient conditions are given for the existence of a finite sequence of gains for which every initial state can be driven to the origin. For a one input, one output system, the question of what terminal states can be reached from a given initial state is resolved. It is shown that an important ingredient for these problems is the semigroup of integers generated by the set{k:cA^{k-1}b neq 0, 1 leq k leq k leq 3n}(for a single input, single output system of dimensionn). It is also natural to use a pair of "canonical forms," in the guise of polynomials, to represent states. One is useful for input considerations and the other for output considerations. For output feedback problems one must further distinguish between two polynomials which are equivalent in the sense that they represent the same state. This is due to the fact that some polynomials are ill-conditioned in that they would have us use a nonzero input when the output vanishes.     

Two singular value inequalities and their implications in H∞approach to control system design

In this note we prove that ifAandBare both nonnegative definite Hermitian matrices andA - Bis also nonnegative definite, then the singular values of A and B satisfy the inequalitiessigma_{i}(A)geq sigma_{i}(B), wherebar{sigma}(cdot) = sigma_{1}(cdot) geq sigma_{2}(cdot) geq '" geq sigma_{m}(cdot) = underbar{sigma}(.)denote the singular values of a matrix. A consequence of this property is that, in a nonsquare H^{infty} optimization problem, ifsup_{omega} bar{sigma}[Z(jsigma)] {underline{underline Delta}} sup_{omega} bar{sigma}[x(jomega)^{T}/ Y(jomega)^{T}]^{T} = lambda, then the singular values ofXandYsatisfy the inequalitylambda^{2} geq max_{i} sup_{omega} [sigma_{i}^{2}(X) + sigma_{m-i-1}^{2}(Y)]wheremis the number of columns of the matrixZ.     

Stability of sampled-data composite systems with many nonlinearities

A sampled-data composite system given by a set of vector difference equationsx_{i}(tau + 1) - x_{i}(tau) = sum min{j = 1} max{n} A_{ij} f_{j}[x_{j}(tau)], i = 1 ..., nis dealt with. The system given byx_{i}(tau + 1) - x_{i}(tau) = A_{ij} f_{i}[x_{i}(tau)]is referred to as theith isolated subsystem. It is shown that the composite system is asymptotically stable in the large if the fisatisfy certain conditions and the leading principal minors of the determinant|b_{ij}|, i,j = 1, ..., n,are all positive. Here, the diagonal element biiis a positive number such that|x_{i}(tau + 1)| - |x_{i}(tau) | leq - b_{ij}| f_{i}[x_{i}(tau)]|holds with regard to the motion of theith isolated subsystem, and the nondiagonal elementb_{ij} , i neq j, is the minus of|A_{ij}|, which is defined as the maximum of|A_{ij}x_{j}|, for|x_{j}| = 1. Some extensions of this result are also given. Composite relay controlled systems are studied as examples.     

Stability conditions for annth-order nonlinear time-varying differential system

The stability of a system described by thenth-order differential equationy^(n) + a_{n-1}Y^(n-1) + ... + a_{1}dot{Y} + a_{0}y = 0wherea_{i} = a_{i}(t, y, dot{y}, ... , y^(n-1)),i = 0,1,2, ... , n-1is considered. It is shown that if the instantaneous roots of the characteristic equation of the system are always contained in a circle on the complex plane with center (- z, 0),z > 0and radius ω such thatfrac{z}{Omega} > {{1, n = 1}{sqrt{2n(n-1)}, n geq 2}then the system is uniformly asymptotically stable in the sense of Liapunov.     

Optimum control of linear systems with a "Modified" energy constraint

Linear control processes are considered under the following optimization criteria: 1) minimization of the terminal error and 2) minimization of the required time (T) to reach a desirable state. The constraint on the control vector(u(t))is considered to beintliminf{t_{j}} limsup{t_{j+1}} u'(t)Qu(t)dt leq c_{j} j = 1, 2, ... mwhereQis a positive definite matrix, andt_{j} < t represents any given interval contained in the interval0, and cjcan be considered as the available energy during that interval. A condition of optimality has been obtained which can be used analytically. Furthermore, a numerical procedure is developed for determination of the optimum control vector.     

On the global controllability of perturbed controllable systems

By using a recent theorem of Davison and Kunze [1], it is shown that, if certain conditions hold such that the systemdot{x} = A(x,t)x + B(x,t)uis globally controllable, then the perturbed systemdot{x} = [A(x,t) + epsilontilde{A}(x,t)]x + [B(x,t) + epsilontilde{B}(x,t)u, wheretilde{A}andtilde{B}are bounded, is also globally controllable, provided ε is small enough. In particular, ifdot{x} = A(t)x + B(t)uis controllable, then so is the perturbed systemdot{x} = [A(t) + epsilontilde{A}(x,t)]x + [B(t) + epsilontilde{B}(x,t)]u.     

The stability of an nth-order nonlinear time-varying differential system

The stability of a system described by annth order differential equationy^{(n)} + a_{n-1}y^{(n-1)} + . . . + a_{1}y + a_{0} = 0wherea_{i}=a_{i}(t, y, dot{y}, . . . , y^{(n-1)}), i=0, 1, . . . , n - 1, is considered. It is shown that if the roots of the characteristic equation of the system are always contained in a circle on the complex plane with center(-z, 0), z > 0, and radius Ω such thatfrac{z}{Omega} > 1 + nC_{[n/2]}where[n/2]= nearest integergeq n/2andnC_{m} = n!/m!(n-m)!, wherenandmare integers, then the system is uniformly asymptotically stable in the sense of Liapunov.     

A homotopy method for eigenvalue assignment using decentralized state feedback

This paper addresses the following problem. Given an interconnected systemMcomposed ofNsubsystems of the formA_{i} + B_{i}K_{i},i = 1,..., N , (A_{i}, B_{i}), a controllable pair, and where the off diagonal blocks ofMlie in the image of the appropriate Bi, then is it possible to arbitrarily assign the characteristic polynomial ofMby a suitable selection of the characteristic polynomials ofA_{i} + B_{i}K_{i}? Moreover, is it possible to compute the appropriate characteristic polynomials of theA_{i} + B_{i}K_{i}(or equivalently construct the Ki) needed to do so? The first question is answered by constructing a mappingF: R^{n} rightarrow R^{n}which maps a prescribed set ofnof the feedback gains (elements ofK_{i}, i=1,...,N) to thencoefficients of the characteristic polynomial ofM. The question then becomes, given ap in R^{n}, doesF(x) = phave a solution? The answer is found by constructing a homotopyH: R^{n}x[O.1] rightarrow R^{n}whereH(x,1)= F(x)andH(x,0)is some "trivial" function. Degree theory is then applied to guarantee that there exists anx(t)such thatH(x(t), t) = pfor alltin [0,1]. The parameterized Sard's theorem is then utilized to prove that (with probability 1)x(t)is a "smooth" curve, and hence can be followed numerically fromx(0)tox(1)by the solution of a differential equation (Davidenko's method).     

On the identification of state-derivative-coupled systems

This short paper treats one aspect of the identification of state-derivative-coupled systems, such asMdot{x}(t) = Ax(t) + Bu(t) + w(t)whereM neq I, andMis invertible. This equation can also be written asdot{x}(t) = F_{1}x(t) + F_{2}u(t) + omega(t). We assume that reduced form parameters (F_{1}, F_{2}) are identifiable and develop a sequence of tests for establishing the identifiability of structural parameters (M, A, B) from (F_{1} F_{2}). The tests are constructive, in that they not only can be used to ascertain the identifiability of (M, A, B); but, if (M, A, B) are not identifiable, can also indicate corrective actions to be taken so that (M, A, B) are identifiable.     

Control of decentralized systems with distributed controller complexity

An approach is presented which allows design of stabilizing decentralized controllers for linear systems with two scalar channels, such that each local controller is endowed with some dynamics while the sum of the orders is kept smaller than the system order. It is shown that for almost allnth-order systems with two scalar channels the local controllers can be chosen such that their orders δ1and δ2satisfydelta_{1} +delta_{2}leq n - 2, max{delta_{1}, delta_{2}} leq max {(n - 1)/2, n -1 - (mu_{0}/2)}, whereμ0is the number of stable zeros of the cross Coupling transfer functions in the system. The approach is to design one of the local controllers such that the McMillan degree of the resulting one-channel system is reduced. Then the other local controller only has to deal with a model of reduced dimension and can thus be chosen of lower order.     

Singular systems: A new approach in the time domain

A new approach in the time domain is developed for the study of singular linear systems of the formEdot{x} = Ax + Bu, y = CxwithEsingular. Central to the approach is the fundamental triple((alpha E - A)^{-1}E, (alphaE - A)^{-1}B, C)with α a real number satisfying det(alpha E - A) neq 0. Controllability and observability matrices are expressed in terms of the fundamental triple. New tests for impulse controllability and impulse observability are also established. Feedback control problems including pole placement, decoupling, and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form ofu = F(alpha x - dot{x})+ v.